Different Color for region of intersection
The "Boolean Modifier" doesn´t work as expected with non-manifold meshes. When you add a plane use the "Solidify Modifier" first, to give it a slight thickness (e.g. 0.0001).

Give the plane a "Boolean Modifier" afterwards and set the target to the cone. Set "Operation" to "Difference". Copy the plane with ShiftD and press Esc to avoid moving the copy. For the copy set the "Operation" of the "Boolean Modifier" to "Intersect".

Edit: The pictures show the Modifier Tab in the Properties Panel (click the wrench icon). Note that the order of the modifiers in the modifier stack is important.





You could also parent the two planes so they can be moved around together. Hold Shift and select both plane objects with RMB. Press CtrlP and select "Object".

Afterwards materials can be assigned on every object separatly.
You can then use Freestyle for line rendering.

The advantage of using modifiers is fast editing and your objects are easy to animate.